"Who should be my first Advisory clients: existing ones, or new ones?"

In almost all cases,*  start with existing clients. 

Why?  

You have a pre-existing foundation to build on. Your existing clients know, like, and trust you, more than your prospects do. This makes it easier to:
 

• weave in a conversation about your new service and how they might benefit

• pilot your process with them, as you learn on the fly

• add value quickly, because you already know their business, their challenges, and what they can do to improve

...

With existing clients, the process may be new to you, but the business and the relationship are familiar. 

With new clients, the process, business, and relationship are all new to you. 

You already have enough of a learning curve to deal with simply by adding Advisory. 

Make it easier on yourself and offer it starting with existing clients. 

...

*only if you really want to start fresh should you begin with new clients when transitioning to Advisory.